Properties

Label 2-507-13.3-c1-0-6
Degree $2$
Conductor $507$
Sign $0.990 - 0.134i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.02 − 1.77i)2-s + (0.5 + 0.866i)3-s + (−1.09 + 1.90i)4-s − 3.35·5-s + (1.02 − 1.77i)6-s + (1.12 − 1.94i)7-s + 0.405·8-s + (−0.499 + 0.866i)9-s + (3.43 + 5.95i)10-s + (2.46 + 4.27i)11-s − 2.19·12-s − 4.60·14-s + (−1.67 − 2.90i)15-s + (1.78 + 3.08i)16-s + (−0.455 + 0.789i)17-s + 2.04·18-s + ⋯
L(s)  = 1  + (−0.724 − 1.25i)2-s + (0.288 + 0.499i)3-s + (−0.549 + 0.951i)4-s − 1.50·5-s + (0.418 − 0.724i)6-s + (0.424 − 0.735i)7-s + 0.143·8-s + (−0.166 + 0.288i)9-s + (1.08 + 1.88i)10-s + (0.744 + 1.28i)11-s − 0.634·12-s − 1.23·14-s + (−0.433 − 0.750i)15-s + (0.445 + 0.771i)16-s + (−0.110 + 0.191i)17-s + 0.482·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.990 - 0.134i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.990 - 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671220 + 0.0452133i\)
\(L(\frac12)\) \(\approx\) \(0.671220 + 0.0452133i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (1.02 + 1.77i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 3.35T + 5T^{2} \)
7 \( 1 + (-1.12 + 1.94i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.46 - 4.27i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.455 - 0.789i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.90 - 3.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.01 + 1.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.96 - 3.41i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.82T + 31T^{2} \)
37 \( 1 + (-4.40 - 7.62i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.46 - 6.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.14 + 1.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.80T + 47T^{2} \)
53 \( 1 - 0.542T + 53T^{2} \)
59 \( 1 + (2.35 - 4.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.83 - 3.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.760 + 1.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.18 + 2.05i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.41T + 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 + 2.30T + 83T^{2} \)
89 \( 1 + (5.02 + 8.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.06 - 13.9i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.77959086477612137515564685839, −10.23547113594437738078261786039, −9.376616877668326657782812872209, −8.322793297264157429834991597208, −7.81768680488412609086659710634, −6.60840691433536356160434783265, −4.43359241058474516446512670875, −4.12798171446516750194767231604, −2.91006810726655084707858498355, −1.32037328581597356234840187406, 0.56154658524569404100272218791, 2.91388535327318393574000253626, 4.24004163345415151061307309970, 5.68755344814861957342415173623, 6.56305332323564014638949561189, 7.43673493436967250653317489491, 8.289714381814471901917375906605, 8.555180631326469423040534602925, 9.453010777598746035722693181521, 11.11067515884928089367705741191

Graph of the $Z$-function along the critical line